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P.M.R.J.O. Dewilde
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Tree Quasi-Separable matrices
A simultaneous generalization of sequentially and hierarchically semiseparable representations
We present a unification and generalization of what is known in the literature as sequentially and hierarchically semiseparable (SSS and HSS) representations for matrices. These so-called tree quasi-separable (TQS) matrices contain sparse matrices with tree-structured adjacency graphs as an important subcase. TQS matrices inherit all the favorable algebraic properties of SSS and HSS under addition, products, and inversion. To arrive at these properties, we prove a key result that characterizes the conversion of any dense matrix into a TQS representation. Here, we specifically show through an explicit construction that the size of the representation is dictated by the ranks of certain Hankel blocks of the matrix. Analogous to SSS and HSS, TQS matrices admit fast matrix-vector products and direct solvers. A sketch of the associated algorithms is provided.
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We present a unification and generalization of what is known in the literature as sequentially and hierarchically semiseparable (SSS and HSS) representations for matrices. These so-called tree quasi-separable (TQS) matrices contain sparse matrices with tree-structured adjacency graphs as an important subcase. TQS matrices inherit all the favorable algebraic properties of SSS and HSS under addition, products, and inversion. To arrive at these properties, we prove a key result that characterizes the conversion of any dense matrix into a TQS representation. Here, we specifically show through an explicit construction that the size of the representation is dictated by the ranks of certain Hankel blocks of the matrix. Analogous to SSS and HSS, TQS matrices admit fast matrix-vector products and direct solvers. A sketch of the associated algorithms is provided.
This issue of the CAS magazine presents papers that could not be accommodated in the 1st part of the Alfred Fettweis memorial special issue that appeared as the December 2018 issue. While the 1st part provided extensive views of Alfred Fettweis' personal life, scientific contributions, and several papers dealing with areas that were influenced by his scientific and technological contributions at large, many other topics could not be included due to multiple reasons which included primarily lack of space, and also the vast expanse of diverse topics that Fettweis had worked on during his long career - both before and after his formal retirement from the scientific/technical enterprise.
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This issue of the CAS magazine presents papers that could not be accommodated in the 1st part of the Alfred Fettweis memorial special issue that appeared as the December 2018 issue. While the 1st part provided extensive views of Alfred Fettweis' personal life, scientific contributions, and several papers dealing with areas that were influenced by his scientific and technological contributions at large, many other topics could not be included due to multiple reasons which included primarily lack of space, and also the vast expanse of diverse topics that Fettweis had worked on during his long career - both before and after his formal retirement from the scientific/technical enterprise.
Journal article
(2018)
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Patrick Dewilde
Orthogonal filtering is a method to extract essential information from digital data using orthogonal transformations. It belongs to the category of Wave Digital Filters (WDF?s) as originally defined and considered by the late Alfred Fettweis, one of the principal founders of modern digital filter theory. In the original WDF theory, filtering is done using adders and an algebraically minimal number of multipliers exclusively. When, instead, the arithmetic is based (also exclusively) on purely orthogonal transformations (Jacobi/Givens rotations), a much larger category of lossless digital filters is obtained. In this paper, it is shown how central classical problems with many engineering applications, namely quadratically optimal tracking (Bellman), linear least squares estimation (Kalman) and spectral factorization (Wiener), among many other types of filters, produce natural orthogonal filters and can be obtained and designed using nothing more than orthogonal transformations. Simple proofs based on these insights are provided, together with a streamlined realization theory for the resulting data processors and filters. The paper uses nothing more than elementary matrix theory, and should be accessible to students with no other background, although it does at some point make the connection with the Beurling-Lax theory on inner-outer factorization and the Wiener theory on spectral factorization, putting these theories in a purely matrix algebra context.
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Orthogonal filtering is a method to extract essential information from digital data using orthogonal transformations. It belongs to the category of Wave Digital Filters (WDF?s) as originally defined and considered by the late Alfred Fettweis, one of the principal founders of modern digital filter theory. In the original WDF theory, filtering is done using adders and an algebraically minimal number of multipliers exclusively. When, instead, the arithmetic is based (also exclusively) on purely orthogonal transformations (Jacobi/Givens rotations), a much larger category of lossless digital filters is obtained. In this paper, it is shown how central classical problems with many engineering applications, namely quadratically optimal tracking (Bellman), linear least squares estimation (Kalman) and spectral factorization (Wiener), among many other types of filters, produce natural orthogonal filters and can be obtained and designed using nothing more than orthogonal transformations. Simple proofs based on these insights are provided, together with a streamlined realization theory for the resulting data processors and filters. The paper uses nothing more than elementary matrix theory, and should be accessible to students with no other background, although it does at some point make the connection with the Beurling-Lax theory on inner-outer factorization and the Wiener theory on spectral factorization, putting these theories in a purely matrix algebra context.
The main objects of this chapter are “semi-separable systems,” sometimes called “quasi-separable systems.” These are systems of equations, in which the operator has a special structure, called “semi-separable” in this chapter. By this is meant that the operator, although typically infinite dimensional, has a recursive structure determined by sequences of finite matrices, called transition matrices. This type of operator occurs commonly in Dynamical System Theory for systems with a finite dimensional state space and/or in systems that arise from discretization of continuous time and space. They form a natural generalization of finite matrices and a complete theory based on sequences of finite matrices is available for them. The chapter concentrates on the invertibility of such systems: either the computation of inverses when they exist, or the computation of approximate inverses of the Moore–Penrose type when not. Semi-separable systems depend on a single principal variable (often identified with time or a single dimension in space). Although there are several types of semi-separable systems depending on the continuity of that principal variable, the present chapter concentrates on indexed systems (so-called discrete-time systems). This is the most straightforward and most appealing type for an introductory text. The main workhorse is “inner–outer factorization,” a technique that goes back to Hardy space theory and generalizes to any context of nest algebras, as is the one considered here. It is based on the definition of appropriate invariant subspaces in the range and co-range of the operator. It translates to attractive numerical algorithms, such as the celebrated “square-root algorithm,” which uses proven numerically stable operations such as QR-factorization and singular value decomposition (SVD).
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The main objects of this chapter are “semi-separable systems,” sometimes called “quasi-separable systems.” These are systems of equations, in which the operator has a special structure, called “semi-separable” in this chapter. By this is meant that the operator, although typically infinite dimensional, has a recursive structure determined by sequences of finite matrices, called transition matrices. This type of operator occurs commonly in Dynamical System Theory for systems with a finite dimensional state space and/or in systems that arise from discretization of continuous time and space. They form a natural generalization of finite matrices and a complete theory based on sequences of finite matrices is available for them. The chapter concentrates on the invertibility of such systems: either the computation of inverses when they exist, or the computation of approximate inverses of the Moore–Penrose type when not. Semi-separable systems depend on a single principal variable (often identified with time or a single dimension in space). Although there are several types of semi-separable systems depending on the continuity of that principal variable, the present chapter concentrates on indexed systems (so-called discrete-time systems). This is the most straightforward and most appealing type for an introductory text. The main workhorse is “inner–outer factorization,” a technique that goes back to Hardy space theory and generalizes to any context of nest algebras, as is the one considered here. It is based on the definition of appropriate invariant subspaces in the range and co-range of the operator. It translates to attractive numerical algorithms, such as the celebrated “square-root algorithm,” which uses proven numerically stable operations such as QR-factorization and singular value decomposition (SVD).
This paper describes how the Surface Integrated Field Equations method (SIFE) is implemented to solve 3D Electromagnetic (EM) problems on substrates in which high contrast materials occur. It gives an account of the promising results that are obtained with it when compared to traditional approaches. Advantages of the method are the highly improved flexibility and accuracy for a given discretization level, at the cost of higher computational complexity.
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This paper describes how the Surface Integrated Field Equations method (SIFE) is implemented to solve 3D Electromagnetic (EM) problems on substrates in which high contrast materials occur. It gives an account of the promising results that are obtained with it when compared to traditional approaches. Advantages of the method are the highly improved flexibility and accuracy for a given discretization level, at the cost of higher computational complexity.